First-Order System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem

Cai, Zhiqiang; Manteuffel, Thomas A.; McCormick, Stephen F.; Parter, Seymour V.

doi:10.1137/s0036142995294930

Cited by 55 publications

(78 citation statements)

References 15 publications

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“…a) Least-squares mixed method The method was analysed in [21] and, under the name First-Order-System Least-Squares (FOSLS), in [10], [11] (see also the references therein). In this method, the saddle-point (min-max) problem (37)- (38) is reformulated as a quadratic minimization (min-min) problem inf…”

Section: Approximation Of the Mixed Solution In Hmentioning

confidence: 99%

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Repin

Sauter²,

Smolianski³

2007

SIAM J. Numer. Anal.

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The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and the individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As the applications, the diffusion problem as well as the problem of linear elasticity are considered.

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Section: Approximation Of the Mixed Solution In Hmentioning

confidence: 99%

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Repin

Sauter²,

Smolianski³

2007

SIAM J. Numer. Anal.

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“…The (generalized) div-grad decomposition used above can also be used for the linear elasticity equations (1.17)-(1.19); see [37]. The new variables are again V = (grad u) t and are called "displacement fluxes" in [37], put perhaps should be more properly referred to as 'displacement gradients."…”

Section: Div-grad Decompositions Of the Stokes Problemmentioning

confidence: 99%

“…As a result, least-squares finite element methods in these settings are among the best understood, studied, and tested from both the theoretical and computational viewpoints. Our discussion will also include least-squares methods for convection-diffusion and other second-order elliptic problems (see [6], [24], [26], [33], [35], [38], [42]- [45], [48], [52], [62], [71], [72], [75], [86], [95], [106], [107], and [104]), linear elasticity (see [34], [36], and [37]), inviscid, compressible flows (see [61], [64], [67], [99], and [118]), and electromagnetics (see [46], [58], [60], [97], and [116]. )…”

mentioning

confidence: 99%

Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

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“…Cai and co-workers [24][25][26][27][28][29] studied the behaviour of equivalent first-order formulations of secondorder systems and found that FE implementation of such first-order systems yields uniform optimal performance. The higher-order differential equations are transformed to first-order differential equations by introducing new dual variables.…”

Section: First-order Systemsmentioning

confidence: 99%

“…In earlier works of Cai et al [24,25], they developed the theory of first-order system formulation for general second-order elliptic PDEs. This methodology has been then extended to the Stokes equations [26] in two and three dimensions, elasticity problems [27][28][29], and boundary value problems with Robin boundary conditions [30]. However, the efforts have been mainly concentrated in using weak-form Galerkin or weak-form least-squares formulation.…”

Section: Introductionmentioning

confidence: 99%

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

Mai‐Duy

Tran-Cong

et al. 2009

Numerical Meth Engineering

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In this paper, high order systems are reformulated as first order systems which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretisation with 1D-integrated radial basis function networks (1D-IRBFN) [1]. The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates.

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First-Order System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem

Cited by 55 publications

References 15 publications

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Two‐Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Finite Element Methods of Least-Squares Type

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

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